Symmetrized pseudofunction algebras from $L^p$-representations and amenability of locally compact groups

We show via an application of techniques from complex interpolation theory how the $L^p$-pseudofunction algebras of a locally compact group $G$ can be understood as sitting between $L^1(G)$ and $C^{\ast }(G)$. Motivated by this, we collect and review various characterizations of group amenability connected to the $p$-pseudofunction algebra of Herz and generalize these to the symmetrized setting. Along the way, we describe the Banach space dual of the symmetrized pseudofuntion algebras on $G$ associated with representations on reflexive Banach spaces.